Robust optimal estimation of location from discretely sampled functional data

TL;DR

This paper introduces a new class of robust, optimal estimators for the location parameter in discretely sampled functional data, addressing limitations of existing methods in handling anomalies and measurement errors.

Contribution

It proposes the first minimax rate optimal robust estimators based on M-type smoothing splines for discretely sampled functional data, applicable to various observation schemes.

Findings

Estimator is minimax rate optimal under certain conditions.

Demonstrates superior performance in simulations and real Covid-19 data.

Applicable to both common and independent sampling schemes.

Abstract

Estimating location is a central problem in functional data analysis, yet most current estimation procedures either unrealistically assume completely observed trajectories or lack robustness with respect to the many kinds of anomalies one can encounter in the functional setting. To remedy these deficiencies we introduce the first class of optimal robust location estimators based on discretely sampled functional data. The proposed method is based on M-type smoothing spline estimation with repeated measurements and is suitable for both commonly and independently observed trajectories that are subject to measurement error. We show that under suitable assumptions the proposed family of estimators is minimax rate optimal both for commonly and independently observed trajectories and we illustrate its highly competitive performance and practical usefulness in a Monte-Carlo study and a real-data example involving recent Covid-19 data.